In the above applications, for example, transmitter filters are tasked with shaping the transmitted pulse, while receiver filters are tasked with rejecting interfering neighboring channels and reducing noise. The frequency and pulse responses of combined transmitter filters and receiver filters are the product of their frequency responses and the convolution of their pulse responses. In order to eliminate inter symbol interference (ISI) and inter channel interference (ICI), the combined filter characteristics should satisfy the following Nyquist criteria: the signal in the center of the eye should be maximum for the received symbol, while it should be zero for all other symbols associated with the same and neighboring channels. The ISI elimination property is enforced by the central symmetry of the slopes of the power frequency response around the points of half-width-at-half-maximum (HWHM), located at ±half of the baud from the center frequency of the channel. This frequency response corresponds to the pulse response featuring a central main lode and time decaying side lobes, which cross zero exactly at the center of each bit. The ICI elimination property is enforced by the steep slopes of the frequency response, amounting to zero outside of the channel bandwidth.
The most commonly known example of the Nyquist shaped frequency response is a raised cosine (RC) function having a shaping parameter, α, which is called a roll off or excess bandwidth factor. When α=0, the frequency response is most confined and has the shape of a uniform rectangle between ±half of the baud from the center frequency of the channel. Alternatively, when α=1, the frequency response has the shape of a RC restricted between ±baud from the center frequency of the channel, with intermediate cases having the uniform central portion of the frequency and RC slopes. It is important to note that, for a QAM format, the ICI is completely eliminated only for α=0, while, for larger roll offs, the offset QAM (OQAM) format is more appropriate.
Matched filtering properties of transmitter and receiver Nyquist filters are ensured by equally splitting the combined Nyquist frequency response of the channel between the transmitter filter and the receiver filter. As a result, the exemplary combined RC response is split into two equal square root RC (SRRC) frequency responses. This results in optimal noise reduction within the channel.
Conventionally, transmitter and receiver filters are realized by digital signal processing (DSP) means as linear phase digital finite impulse response (FIR) SRRC filters.
DSP chips used for signal pre and post filtering and equalization, although immensely popular, tend to become larger and more power hungry with an increase in the bandwidth of the processed signals. Analog filters are able to perform the same tasks faster, have smaller size, consume less power, and tend to use larger and cheaper semiconductor process geometries.
There are known filter designs, and even commercially available filter products, based on lumped inductor-capacitor (LC) components that feature both RC and SRRC responses. In these designs and products, the challenging tasks of the synthesis and realization of an SRRC response with a linear phase are accomplished. However, they have limited bandwidth, are bulky and complex, and have very high tolerance requirements for their constituent reactive elements, which also involve mutual inductances in addition to the regular inductors and capacitors.
An analog FIR filter may be implemented most straightforwardly as a continuous time transmission line based filter, wherein each transmission line (typically implemented as a stripline) serves as an analog delay element. The primary disadvantage of this type of filter is the large physical size of each transmission line, resulting in severe area inefficiency. As a result, it is difficult to implement designs of this type in integrated circuit technologies, especially for low baud signals that would require correspondingly long delay lines.
Transmission lines in the above type of filter may be realized in the form of balanced LC ladder based analog delay elements, instead of striplines, making them much more compact and integratable, although still quite sizable due to the many inductor coils necessary for the realization of the required delays. The primary disadvantage of this type of filter is the signal distortion caused by the non-ideal phase characteristics of the LC ladder based analog delay elements.
It is possible to eliminate the designated analog delay elements of the FIR filter by pipelining the analog voltage waveforms along the chain of varying taps from one repeating amplifier to another. The necessary delays are realized as inherent delays of the repeating amplifiers, which are designed to have the Bessel type transfer characteristics with fixed and flat delay over the bandwidth of the signal delay. The primary disadvantage of this type of filter is increased complexity and power consumption due to the numerous active analog delay elements, as well as signal distortion and noise accumulation along the chain of amplifiers.
Finally, there is a class of discrete time analog FIR filters, which, according to their designating name, pipeline the discrete time samples of the analog voltage waveforms along the chain of varying taps from one sample-and-hold (S/H) or track-and-hold (T/H) amplifier to another. In this case, the necessary delays are produced by clocking the S/H or T/H amplifiers at time intervals equal to the delay time increments between the taps of the FIR filter. Such designs are disadvantaged by the need for fast and complex sampling circuitry, and suffer from noise, offset, and nonlinearity errors accumulating as the signal propagates along the chain of S/H or T/H stages.